This may be too late for you, but you could consider parking in the Business Parking T1-3 (formerley Purple Parking) on the northside peri road and then walk to the T5 long stay transfer bus, which is next door to it (but may be a 5-10 min walk to get from car to bus pickup.)

I’m not aware of a on airport car park that serves both T1 and T5.

Maybe another option is off airport parking, with drop off and pick up from different terminals?

End of this story, I landed back into T1 from Frankfurt. Was in T5 Business Parking, using Train a couple of lifts and the buggy transfer, within 25 minutes of landing.. not bad!

As a side issue, the LBC on Friday was hot with complaints about the road closures in London with the bike day.

Getting to the airport in Frankfurt this morning, taxi fare was Euros 51, instead of Euros 24 – due to a similar bike event in Frankfurt with a lot of traffic being clueless about the event and no diversion signs evident.